Optimal code rates for the Lorentzian channel: Shannon codes and LDPC codes

In this paper, we take an information-theoretic approach to obtaining optimal code rates for error-control codes on a magnetic storage channel approximated by the Lorentzian channel. Code rate optimality is in the sense of maximizing the information-theoretic user density along a track. To arrive at such results, we compute the achievable information rates for the Lorentzian channel as a function of signal-to-noise ratio and channel density, and then use these information rate calculations to obtain optimal code rates and maximal linear user densities. We call such (hypothetical) optimal codes "Shannon codes" We then examine optimal code rates on a Lorentzian channel assuming low-density parity-check (LDPC) codes instead of Shannon codes. We employ as our tool extrinsic information transfer (EXIT) charts, which provide a simple way of determining the capacity limit (or decoding threshold) for an LDPC code. We demonstrate that the optimal rates for LDPC codes coincide with those of Shannon codes and, more important, that LDPC codes,are essentially capacity-achieving codes on the Lorentzian channel. Finally, we use the above results to estimate the optimal bit-aspect ratio, where optimality is in the sense of maximizing areal density.